Theorem ax12from12o 2156

Description: Derive ax-12 1925 from ax-12o 2142 and other older axioms.

This proof uses newer axioms ax-5 1557 and ax-9 1654, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-5o 2136 and ax-9o 2138. (Contributed by NM, 21-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax12from12o	⊢ (¬ x = y → (y = z → ∀x y = z))

Proof of Theorem ax12from12o

Step	Hyp	Ref	Expression
1		ax-4 2135	. . . . . 6 ⊢ (∀x x = y → x = y)
2	1	con3i 127	. . . . 5 ⊢ (¬ x = y → ¬ ∀x x = y)
3	2	adantr 451	. . . 4 ⊢ ((¬ x = y ∧ y = z) → ¬ ∀x x = y)
4		equtrr 1683	. . . . . . . 8 ⊢ (z = y → (x = z → x = y))
5	4	equcoms 1681	. . . . . . 7 ⊢ (y = z → (x = z → x = y))
6	5	con3rr3 128	. . . . . 6 ⊢ (¬ x = y → (y = z → ¬ x = z))
7	6	imp 418	. . . . 5 ⊢ ((¬ x = y ∧ y = z) → ¬ x = z)
8		ax-4 2135	. . . . 5 ⊢ (∀x x = z → x = z)
9	7, 8	nsyl 113	. . . 4 ⊢ ((¬ x = y ∧ y = z) → ¬ ∀x x = z)
10		ax-12o 2142	. . . 4 ⊢ (¬ ∀x x = y → (¬ ∀x x = z → (y = z → ∀x y = z)))
11	3, 9, 10	sylc 56	. . 3 ⊢ ((¬ x = y ∧ y = z) → (y = z → ∀x y = z))
12	11	ex 423	. 2 ⊢ (¬ x = y → (y = z → (y = z → ∀x y = z)))
13	12	pm2.43d 44	1 ⊢ (¬ x = y → (y = z → ∀x y = z))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-4 2135  ax-12o 2142

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by: (None)